Mathematical headaches? Problem solved!
Hi, I'm Colin, and I'm here to help you make sense of maths
Mathematical headaches? Problem solved!
Hi, I'm Colin, and I'm here to help you make sense of maths
Written by Colin+ in ninja maths.
"So," said the Mathematical Ninja, "we meet again."
"In fairness," said the student, "this is our regularly-scheduled appointment."
The Mathematical Ninja was unable to deny this. Instead, it was time for a demand: "Tell me the square root of 22."
"Gosh," said the student. "Between four-and-a-half and five, definitely. 4.7 or so?"
The Mathematical Ninja nodded. "Four and seven tenths is an excellent initial approximation. Allow me to demonstrate a better one."
The student allowed.
"You have correctly - I presume - made your initial estimate by noting that 22 is 3 smaller than $5^2$, so $5 - \frac{3}{2\times 5}$ is close to the root."
"Of course, sensei," said the student, who had done nothing of the sort. He had interpolated quickly.
"A better second estimate is $5 - \frac{30}{97}$. One gets that - in this case, with the key numbers 3 and 10 and a previous estimate of $\frac{a}{b}$, as $\frac{3b}{10b+a}$."
"Thirty... over 100 + 3. Wai... wah!"
"$a$ is -3, remember."
The student was unlikely to forget that now. "So, I could apply the same process again to get a correction of..." (pause for thought) " $\frac{291}{940}$?"
The Mathematical Ninja nodded again. "You could continue the process for as long as you felt the need - each iteration gets you a slightly better estimate. $5-\frac{291}{940}$ is correct to one part in half a million."
The student felt that there was little need to make it any more accurate than that - although he'd never tell the Mathematical Ninja that. He had developed a little sense.
elmer
I don’t get this part ‘so 5 – 3/2×5 is close to the root’
Colin
That’s one of the Mathematical Ninja’s favourite tricks!
You start from $\sqrt{22} = \sqrt{25-3} = 5 \sqrt{1 – \frac{3}{25}}$, then apply the binomial expansion to the square root: $\left(1 + x\right)^{\frac{1}{2}} \approx 1 + \frac{1}{2}x + …$. That gives $5 \left(1 – \frac{3}{2\times 25}\right) = 5 – \frac{3}{2 \times 5}$.
Generally, a fairly good approximation for a square root in the form $\sqrt{a^2 + b}$, with $b < a$, is $a + \frac{b}{2a}$. Hope that helps!